Convergence and correctness of belief propagation for weighted min–max flow
In this paper, we investigate the performance of message-passing algorithms for the weighted min–max flow (WMMF) problem which was introduced by Ichimori et al. (1980). WMMF was well studied in combinational optimization, as it provides important applications in time transportation problem and the s...
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| Vydané v: | Discrete Applied Mathematics Ročník 354; s. 122 - 130 |
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Elsevier B.V
15.09.2024
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| Abstract | In this paper, we investigate the performance of message-passing algorithms for the weighted min–max flow (WMMF) problem which was introduced by Ichimori et al. (1980). WMMF was well studied in combinational optimization, as it provides important applications in time transportation problem and the storage management problem. We develop a message-passing algorithm called min–max belief propagation (BP) for determining the optimal solution of WMMF. As the main result of this paper, we prove that for a digraph of size n, BP converges to the optimal solution within O(n3) time after O(n) iterations if the optimal solution of the underlying min–max flow problem instance is unique. To the best of our knowledge, the fastest polynomial time algorithm for WMMF runs in essentially O(n6) time among the known algorithms, where n is the number of vertices. On the other hand, it is one of a very few instances where BP is proved correct with fully-polynomial running time. |
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| AbstractList | In this paper, we investigate the performance of message-passing algorithms for the weighted min–max flow (WMMF) problem which was introduced by Ichimori et al. (1980). WMMF was well studied in combinational optimization, as it provides important applications in time transportation problem and the storage management problem. We develop a message-passing algorithm called min–max belief propagation (BP) for determining the optimal solution of WMMF. As the main result of this paper, we prove that for a digraph of size n, BP converges to the optimal solution within O(n3) time after O(n) iterations if the optimal solution of the underlying min–max flow problem instance is unique. To the best of our knowledge, the fastest polynomial time algorithm for WMMF runs in essentially O(n6) time among the known algorithms, where n is the number of vertices. On the other hand, it is one of a very few instances where BP is proved correct with fully-polynomial running time. |
| Author | Zhang, Zan-Bo Gutin, Gregory Guo, Longkun Zhang, Xiaoyan Dai, Guowei |
| Author_xml | – sequence: 1 givenname: Guowei surname: Dai fullname: Dai, Guowei email: guoweidai@njnu.edu.cn organization: School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing, China – sequence: 2 givenname: Longkun surname: Guo fullname: Guo, Longkun email: longkun.guo@qlu.edu.cn organization: Department of Computer Science, Qilu University of Technology, Jinan, China – sequence: 3 givenname: Gregory surname: Gutin fullname: Gutin, Gregory email: gutin@cs.rhul.ac.uk organization: Department of Computer Science, Royal Holloway University of London, Egham, UK – sequence: 4 givenname: Xiaoyan orcidid: 0000-0001-8563-4958 surname: Zhang fullname: Zhang, Xiaoyan email: zhangxiaoyan@njnu.edu.cn organization: School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing, China – sequence: 5 givenname: Zan-Bo orcidid: 0000-0002-0851-4984 surname: Zhang fullname: Zhang, Zan-Bo email: zanbozhang@gdufe.edu.cn organization: School of Statistics & Mathematics, and Institute of Artificial Intelligence & Deep Learning, Guangdong University of Finance & Economics, Guangzhou, China |
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| Cites_doi | 10.1287/opre.1110.1025 10.1126/science.1086309 10.1109/TIT.2011.2110170 10.1016/0012-365X(78)90055-9 10.1109/18.910577 10.1126/science.1073287 10.1137/090753115 10.1109/TIT.2005.850085 10.1287/trsc.25.4.314 10.1109/TIT.2007.915695 10.1109/TIT.2009.2030448 10.1002/net.3230090405 10.1016/0167-6377(86)90079-9 10.1007/s10898-019-00749-2 10.1109/18.910585 10.1126/science.1136800 |
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| Keywords | Min–max BP algorithm Message-passing algorithm Belief propagation Min–max flow |
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| SubjectTerms | Belief propagation Message-passing algorithm Min–max BP algorithm Min–max flow |
| Title | Convergence and correctness of belief propagation for weighted min–max flow |
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